1. Algebra1 Slopes of Parallel and Perpendicular Lines
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2. contains the given point.
Warm Up
Write an equation in slope-intercept form for the line with the given slope that
contains the given point.
1) slope = -5; (2, 4)
2) slope = 0; (-3, 3)
2) y - 4= -2(x -2)
1) y= 3x -4
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3. Parallel Lines
To sell at a particular farmers’ market for a year, there is a $100 membership fee. Then you pay $3 for each hour that you sell at the market. However, if you were a member the previous year, the membership fee is reduced to $50.
• The red line shows the total cost if you are a new member.
• The blue line shows the total cost if you are a returning member.
These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect.
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4. Parallel Lines WORDS GRAPH
Two different non vertical lines are parallel if and only if they have the same slope.
All different vertical lines are parallel.
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5. Identifying Parallel Lines
Identify which lines are parallel.
A) y = 4 x + 3; y = 2; y = 4 x - 5; y = -3
The lines described by y = and y = x - 5 both have slope These lines are parallel. The lines described by y = 2 and y = -3 both have slope 0. These lines are parallel. 4 3
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6. Write all equations in slope-intercept form to determine the slopes.
B) y = 3x + 2; y = -1 x + 4; x + 2y = -4; y - 5 = 3 (x - 1) 2
Write all equations in slope-intercept form to determine the slopes.
y = - x + 4 1 2
y = 3x + 2
slope-intercept form
slope-intercept form
x + 2y = -4
y - 5 = 3 (x - 1)
y - 5 = 3x - 3
2y = -x - 4
2y = -x - 4
y = 3x + 2 2 2
y = - x + 4 1 2
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7. The lines described by y = 3x + 2 and y - 5 = 3(x - 1) have the same slope, but they are not parallel lines. They are the same line. The lines described by y = -1x and x + 2y = -4 represent parallel lines They each have slope -1. 2
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8. Identify which lines are parallel.
Now you try!
Identify which lines are parallel.
1a) y = 2x + 2; y = 2x + 1; y = -4; x = 1
1b) y =3 x + 8; -3x + 4y = 32; y = 3x; y - 1 = 3 (x + 2) 4
1a) y= 2x + 2; y= 2x + 1
1b) y= 3x; y – 1 = 3(x + 2)
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9. Show that ABCD is a parallelogram.
Geometry Application
Show that ABCD is a parallelogram.
Use the ordered pairs and the slope formula to find the slopes of AB and CD .
7 - 5 _= 2
4 – (-1) 5
slope of AB =
3 - 1 _= 2
4 – (-1) 5
slope of AB =
AB is parallel to CD because they have the same slope.
AD is parallel to BC because they are both vertical.
Therefore, ABCD is a parallelogram because both pairs of opposite sides are parallel.
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10. Now you try!
2) Show that the points A (0, 2) , B (4, 2) , C (1, -3) , and D (-3, -3) are the vertices of a parallelogram.
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11. Perpendicular Lines
Perpendicular lines are lines that intersect to form right angles (90°).
WORDS
GRAPH
Two non vertical lines are
perpendicular if and only if the product of their slopes is -1.
Vertical lines are perpendicular to horizontal lines.
CONFIDENTIAL

12. Identifying Perpendicular Lines
Identify which lines are perpendicular: x = -2; y = 1; y = -4x; y + 2 = 1(x + 1) . 4
The graph described by x = -2 is a vertical line, and the graph described by y = 1 is a horizontal line. These lines are perpendicular.
The slope of the line described by y = -4x is -4. The slope of the line described by y + 2 = 1 (x - 1) is 1. 4
(-4) × 1 = -1 4
These lines are perpendicular because the product of their slopes is -1.
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13. 3) y= -4; x =3; y – 6 = 5x + 4 and y = 1x + 2 5
Now you try!
3) Identify which lines are perpendicular: y = -4; y - 6 = 5 (x + 4) ; x = 3; y = -1x + 2. 2
3) y= -4; x =3; y – 6 = 5x + 4 and y = 1x + 2 5
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14. Geometry Application Show that PQR is a right triangle.
If PQR is a right triangle,
PQ will be perpendicular QR .
3 - 1 _= 2
3 –
slope of PQ =
3 - 0 _= -3
3 –
slope of QR =
PQ is perpendicular to QR because 2 × -3 = -1
Therefore, PQR is a right triangle because it contains a right angle.
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15. Now you try!
4) Show that P (1, 4) , Q (2, 6) , and R (7, 1) are the vertices of a right triangle.
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16. Writing Equations of Parallel and Perpendicular Lines
A) Write an equation in slope-intercept form for the line that passes
through (4, 5) and is parallel to the line described by y = 5x + 10.
Step1: Find the slope of the line.
y = 5x The slope is 5.
The parallel line also has a slope of 5.
Step2: Write the equation in point-slope form.
y - y1 = m (x - x1 ) Use point-slope form.
y - 5 = 5( x - 4) Substitute 5 for m, 4 for x1 , and 5 for y1.
Step3: Write the equation in slope-intercept form.
y - 5 = 5 (x - 4)
y - 5 = 5x – Distribute 5 on the right side.
y = 5x Add 5 to both sides.
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17. Step1: Find the slope of the line. y = 3x - 1 The slope is 3.
B) Write an equation in slope-intercept form for the line that passes through (3, 2) and is perpendicular to the line described by y = 3x - 1.
Step1: Find the slope of the line.
y = 3x The slope is 3.
The perpendicular line has a slope of -1 because 3 × -1= -1.
Step2: Write the equation in point-slope form.
y - y1 = m (x - x1 ) Use point-slope form.
y - 2 = -1( x - 3) Substitute -1 for m, 3 for x1 , and 2 for y1. 3
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18. Step3: Write the equation in slope-intercept form.
y - 2 = -1 (x - 3)
y - 2 = -1x Distribute -1 on the right side.
y = -1x Add 2 to both sides. 3
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19. Now you try!
5a) Write an equation in slope-intercept form for the line that passes through (5, 7) and is parallel to the line described by y = 4 x - 6. 5
5b) Write an equation in slope-intercept form for the line that passes through (-5, 3) and is perpendicular to the line described by y = 5x.
5a) y = 4x + 3 ; y = -1x + 2
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20. Identify which lines are parallel.
Assessment
Identify which lines are parallel.
1) y = 6; y = 6x + 5; y = 6x - 7; y = -8
2) y = 3x - 1; y = -2x; y - 3 = 3(x - 5) ; y - 4 = -2(x + 2)
1) y = 6x + 5 and y = 6x – 7; y = 6 and y = -8
2) y = 3x – 1 and y - 3 = 3(x - 5)
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21. 3) Slope of AD= -3 Slope of BC= -3 . AD is ll BC. 2 2
3) Show that ABCD is a trapezoid. (Hint: In a trapezoid, exactly one pair of opposite sides is parallel.)
3) Slope of AD= -3 Slope of BC= -3 . AD is ll BC.
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22. Identify which lines are perpendicular:
4) y = 2x - 4; y = -3 x + 2
Identify which lines are perpendicular:
4) y = 2x - 4; y = -3 x + 2; y = -1; x = 3 3
5) y = -3 x - 4; y - 4 = -7(x + 2) ;
y - 1 = 1(x - 4) ; y - 7 = 7(x - 3) 7 3
5) y = -3 x - 4; y - 7 = 7(x - 3) ;
y - 1 = 1(x - 4) ; y - 4 = -7(x + 2) 7 3
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23. Write an equation in slope-intercept form for the line that is parallel to the given line and that passes through the given point.
6) y = 3x - 7; (0, 4)
7) y = 1x + 5; (4, -3) 2
7) y = 1x – 5 2
6) y = 3x + 4
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24. 8) Write an equation in slope-intercept form for the line that passes through (5, 0) and is perpendicular to the line described by y = -5x + 6. 2
8) y = 2x - 2 5
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25. Parallel Lines Let’s review
To sell at a particular farmers’ market for a year, there is a $100 membership fee. Then you pay $3 for each hour that you sell at the market. However, if you were a member the previous year, the membership fee is reduced to $50.
• The red line shows the total cost if you are a new member.
• The blue line shows the total cost if you are a returning member.
These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect.
CONFIDENTIAL

26. Parallel Lines WORDS GRAPH
Two different non vertical lines are parallel if and only if they have the same slope.
All different vertical lines are parallel.
CONFIDENTIAL

27. Identifying Parallel Lines
Identify which lines are parallel.
A) y = 4 x + 3; y = 2; y = 4 x - 5; y = -3
The lines described by y = and y = x - 5 both have slope These lines are parallel. The lines described by y = 2 and y = -3 both have slope 0. These lines are parallel. 4 3
CONFIDENTIAL

28. Show that ABCD is a parallelogram.
Geometry Application
Show that ABCD is a parallelogram.
Use the ordered pairs and the slope formula to find the slopes of AB and CD .
7 - 5 _= 2
4 – (-1) 5
slope of AB =
3 - 1 _= 2
4 – (-1) 5
slope of AB =
AB is parallel to CD because they have the same slope.
AD is parallel to BC because they are both vertical.
Therefore, ABCD is a parallelogram because both pairs of opposite sides are parallel.
CONFIDENTIAL

29. Perpendicular Lines
Perpendicular lines are lines that intersect to form right angles (90°).
WORDS
GRAPH
Two non vertical lines are
perpendicular if and only if the product of their slopes is -1.
Vertical lines are perpendicular to horizontal lines.
CONFIDENTIAL

30. Identifying Perpendicular Lines
Identify which lines are perpendicular: x = -2; y = 1; y = -4x; y + 2 = 1(x + 1) . 4
The graph described by x = -2 is a vertical line, and the graph described by y = 1 is a horizontal line. These lines are perpendicular.
The slope of the line described by y = -4x is -4. The slope of the line described by y + 2 = 1 (x - 1) is 1. 4
(-4) × 1 = -1 4
These lines are perpendicular because the product of their slopes is -1.
CONFIDENTIAL

31. Geometry Application Show that PQR is a right triangle.
If PQR is a right triangle,
PQ will be perpendicular QR .
3 - 1 _= 2
3 –
slope of PQ =
3 - 0 _= -3
3 –
slope of QR =
PQ is perpendicular to QR because 2 × -3 = -1
Therefore, PQR is a right triangle because it contains a right angle.
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32. You did a great job today!
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